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The measurement of intensities in the Rotation Electron Diffraction Technique
Accepted Manuscript Title: The measurement of intensities in the Rotation Electron Diffraction Technique Authors: M. Buxhuku, V. Hansen, J. Gjønnes PII: DOI: Reference:
S0968-4328(17)30033-1 http://dx.doi.org/doi:10.1016/j.micron.2017.06.006 JMIC 2443
To appear in:
Micron
Received date: Revised date: Accepted date:
27-1-2017 9-6-2017 14-6-2017
Please cite this article as: Buxhuku, M., Hansen, V., Gjønnes, J., The measurement of intensities in the Rotation Electron Diffraction Technique.Micron http://dx.doi.org/10.1016/j.micron.2017.06.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
The measurement of intensities in the Rotation Electron Diffraction Technique M. Buxhuku1, V. Hansen1 and J. Gjønnes2 1Department
of Mechanical and Structural Engineering and Material
Science, University of Stavanger, N-4036 Stavanger, Norway 2Department of Physics, University of Oslo, Gaustadalleen 21, N-0371 Oslo,
Norway
Highlits
Intensity data are recorded as a set of frames during stepwise rotation about an arbitrary axis. A precise description of the diffraction geometry throughout the rotation movement has been given. The defined rotation axis was the basic element to extract the intensity data and to calculate the excitation error. To develop the RED method, a precise description of the diffraction geometry throughout the rotation movement has been given. Reflections, which do not reach the maximum intensity, will not be included in the structure solution and refinement strategy
1
ABSTRACT The Rotation Electron Diffraction Technique, recently developed at the University of Stockholm, has been used to acquire three-dimensional electron diffraction data. A mathematical expression to calculate the excitation errors, 𝑠𝑔 , is suggested by considering the diffraction geometry and the rotation axis. In order to plot the rocking curves, diffraction patterns taken from CoP3 cubic structure (a = 7.708Å, space group Im3̅) are used as examples. Intensities of some reflections are derived by Multigauge software during the beam tilt series. Rocking curves for two Friedel pair reflections with different angles from the rotation axis are plotted. Keywords: Rotation axis, Excitation error, Kikuchi lines, Rotation Electron Diffraction Technique INTRODUCTION The successes of X-ray and neutron diffraction methods in structure determination are rooted in the concept of integrated intensities. The intensities, 𝐼ℎ𝑘𝑙 , thus recorded can be taken to represent the square of the 2
structure amplitude, |𝐹ℎ𝑘𝑙 |2 , apart from correction factors. It has taken a long time to establish techniques for electron diffraction that can produce useful and robust experimental intensities (Weirich et al., 1996; Andersen et al., 1998; Zou et al., 1993; Dorset, 1995). This is due to the strong interactions between the electron beam and the atoms in the crystal, which lead to strong dynamical scattering effects. This is especially so in the most common type of electron diffraction patterns, recorded in Selected Area Electron Diffraction (SAED) or micro-diffraction modes, with the incident beam along a major zone axis. Efforts have therefore been made to establish diffraction techniques where multi beams’ dynamical scattering is suppressed to the extent that the recorded intensity data can be used in the structure solution. A basic feature has been to tilt the incident beam off the main zone axis, reducing the number of simultaneously excited beams. The Precession Electron Diffraction technique (PED), invented by Vincent and Midgley (Vincent & Midgley, 1994), is seen as emulating the precession photography in X-ray crystallography (Buerger, 1964). This is achieved by simultaneous precession of the incident beam and the diffracted beams. In
3
practice only the first layer is used; three-dimensional intensity data must be obtained by a combination of several axes (Gjønnes et al., 1998). The recently developed Rotation Electron Diffraction method (RED), employs a different principle (Mugnaioli et al., 2009; Zhang et al., 2010), that might be compared with X-ray single crystal diffractometry. Continuous three-dimensional diffraction intensities are recorded by a combination of a single tilt holder and beam tilt about a common arbitrary axis. Intensity data are available as a set of frames corresponding to the planar section of the reciprocal space. The recorded intensity data are extracted by a computer program, EDT Collect, as maxima or by integrations over several planar sections of reciprocal space. So far, maximum intensities have been interpreted kinematically in structure solution by this technique (Mugnaioli et al., 2009) but with poorer R values. There is thus a need for improvement of the technique and its application in structure solution. The possibility of including dynamical diffraction in the interpretation, and a more precise description of the experiment remains an important task in the development of the method. 4
A key feature will be a precise specification of the path of the incident beam during the rotation experiment, and thus of the diffraction condition of the excited beams. Two illustrations, referred to as two-dimensional representations are shown: a set of Laue circles in the diffraction spot pattern and a line representing the incident beam in a Kikuchi line pattern.
MATERIALS AND METHODS Two-dimensional representation in the RED experiment Two-dimensional patterns, recorded with the incident electron beam along major zone axes, are standard representations in electron diffraction (see Fig. 1). However, recent developments in quantitative electron diffraction crystallography focus on patterns that are recorded with off-axis incident beam direction. In the rotation electron diffraction methods (RED) (Zhang 5
et al., 2010; Kolb et al., 2007; Kolb et al., 2008), three-dimensional electron diffraction data are collected by rotation about an arbitrary axis, by a combination of goniometer tilt motion and beam tilt. For use in structure studies, each detailed analysis of diffraction geometry during the rotation experiment is essential, viz. of the incident beam direction 𝑘0 and the excitation 𝑠𝑔 of diffracted beams. As a basis for the analysis in two steps, we propose a two-dimensional representation within a beam tilt range of a few degrees rotation. We assume in this analysis that a basic structure is known. As the first step for the chosen range, a common normal 𝑍 is selected, i.e. with crystallographic indices 𝑢𝑣𝑤, which will define a zero layer (ZOLZ) of reflections: hkl – and a starting value 𝑘0 (0). A preliminary rotation direction 𝑌 is estimated from recorded patterns, e.g. with methods explained by Buxhuku et al. (Buxhuku et al., 2013). The vector product, 𝑻 = 𝒀𝑥𝒁
6
will define the direction of the change in the incident wave vector as a function of the rotation angle 𝛽: 𝒌0 (𝛽) = 𝒌0 (0) + 𝑛𝛽 𝑻
(1)
where n is a normalization factor. Once 𝒌𝟎 (𝛽) is given, the excitation errors of reflections 𝑔 can be determined from the standard expression 2𝑘𝑠𝑔 = −𝒈𝟐 − 2𝒌𝒈
(2)
For a given rotation axis and incident beam, a calculation of expected beam tilt angles for those reflections in Bragg condition is carried out. The calculated value of 𝛽(𝑔), for 𝒔𝒈 = 0, must be compared with those obtained from the experimental values collected by the RED method, e.g. by a refinement procedure. Diffraction conditions during the RED experiment are visualized in two different ways:
7
In the diffraction pattern by a set of Laue circles in ZOLZ, excitation errors are defined by the distance of the reflections from the circle (see Fig. 2). In the Kikuchi line diagram, which maps the locations of Bragg conditions, 𝑠𝑔 = 0, where 𝒌𝟎 (𝛽) is represented as a line, excitation errors are again represented by a distance from the corresponding Kikuchi line. Beam tilt represented by Kikuchi line patterns We consider the Kikuchi line diagram of [317] zone axis in the crystal (see Fig. 3). The diffraction patterns show that the ℎ − 3ℎ 0 systematic row is close to the rotation axis (Buxhuku et al., 2013), with 1̅30 reflection almost in Bragg condition. The trace of the incident beam, 𝑘0 , during the beam tilt series is represented by a line in the beam diagram: the beam tilt path. When 𝑘0 is at the position of the Kikuchi line’s reflection e.g. 5̅12, the reflection will be in Bragg condition. These diffraction conditions are also shown in Fig. 2 with a Laue circle center at the point a. The Kikuchi diagram representation is well suited to carry out the indexes of the reflections’ excitation in almost the same angular range. 8
Simulated Kikuchi line patterns, of the planar section (see Fig.3), during ±0.75° beam tilt are shown considering the three axes, XYZ, where 𝑌 is consider as rotation axis Y= 13̅0; 𝑍 is the direction of incident beam 𝑘0 , and 𝑋 the tangent projection. The direction of the line representing 𝑘0 (𝛽) is given by vector product as shown in Eq. (1).
The expectation position for a reflection, 𝑔, is obtained for 𝑠𝑔 = 0; the value of 𝛽(𝑔) thus obtained must be compared with experimental values. If the reflection has a 𝑧 − component an additional term will be included. Converting the diffraction patterns into integrated intensities Planar sections of near [317] zone axis have been processed by several stages on Multigauge. Extracted intensities1 of 51̅2̅ and 5̅12 are collected from 35 – 40 diffraction patterns of the same beam tilt series and plotted as
1
The intensities shown on the rocking curve are relative intensities and scaled to maximum 100
9
a function of the beam tilt angle. The strongest intensities of reflections which appeared during data collection are considered as in Bragg conditions. Due to the geometrical or so-called Lorentz factor, some reflections might not reach the maximum intensity. These reflections and the ones on the rotation axis will not be included in the structure solutions and refinements’ strategy. The rotation axis, RED geometry and the beam tilt angles are used to derive the mathematical expression for 𝑠𝑔 (see Eq. (6)). The rocking curves for 51̅2̅ and 5̅12 are plotted as a function of excitation error. The rocking curve for one Friedel pair of reflections is also plotted, with one relative to the other. The curves show interesting features and differences in the maximum intensities of the same excitation error e.g. the maximum intensities for 5̅12 is almost twice higher than 51̅2̅ (see Fig. 5). Maximum intensities have been used and interpreted kinematically, solving a number of structures, but limiting the application for refinement. Reasonable results in terms of structure factors may be associated with integrated intensities collected during beam tilt diffraction pattern. It should be apparent from the present study that the expression for the excitation 10
error, 𝑠𝑔 , derived from the diffraction pattern of a beam tilt series, and the Kikuchi line patterns can be used to determine the refinement strategy. The features on the curves are investigated further by representing the RED experiment by the Kikuchi diagram. The simulated Kikuchi line patterns during beam tilts (see Fig. 6) showed that the rocking curve features of reflections before Bragg conditions are due to the dynamical effects. During the beam tilt series of 5̅12 (see Fig. 7), other reflections, such as 2̅1̅1, 4̅2̅2 , 3̅21, and 1̅30, will appear in Bragg conditions. The 13̅ 0 and 1̅30 reflections are the closest to the rotation axis. Due to the curvature of the Ewald sphere, these two reflections will be in Bragg conditions longer than the other reflections. The simulated Kikuchi line patterns of 51̅2̅ reflection show only 422̅ and 1̅30 reflections occurring in Bragg conditions. It is apparent from the present study that the Kikuchi line patterns can be used to determine the refinement strategy.
11
Integrated intensities should be equal to the area under the rocking curve when the extracted intensities of reflections are plotted against the excitation error. CALCULATIONS Plotting the rocking curves and defining 𝐬𝐠 expression The data set in the RED were performed and collected in the parallel electron beam mode. Diffraction patterns were summed up and the intensities of reflections were collected as maximum. Intensities of the reflections from different diffraction patterns were plotted as a function of the beam tilt angle. The Lorentz factor is proportional to the “time” the reflection spends close to Bragg conditions and inversely proportional to the velocity with which the Ewald sphere passes through the diffraction condition. Hence, reflections close to the rotation tilt axis and/or incident beam, 𝑘0 , will be visible on many planar sections of the reciprocal space. In consideration of the RED geometry, simulated Kikuchi line patterns, the
12
rotation axis and the beam tilt angles, an expression for the excitation errors, 𝑠𝑔 , is suggested. Referring to Fig. 6, the distance D of reflection, 𝑔, from the rotation axis gives the geometry effect during data collection. The distance D of 𝑔 −reflection is: 𝑔 𝐷 = (𝑘0 cos 𝛼 − ⁄2) sin 𝛽
(3)
Choosing 𝑌 as rotation axis along 𝑍, the optical axis, and 𝑋 perpendicular to the rotation axis and optical axis, the beam in the 𝑋𝑍 plane and 𝑌 component of 𝑔 is irrelevant for the excitation error. In practical terms, the calculation of 𝑠𝑔 for the experimental intensity profiles is: 2𝐾 = −𝟐𝒌𝒈 − 𝑔2 = −2𝑘[𝑔𝑥 + sin 𝛼 𝑔𝑧 cos 𝛼]𝑔2
(4)
13
The component of k0 along the Y axis is 𝑘0 sin 𝛼, ∆(sin 𝛼) ≈ ∆𝛼 and ∆(cos 𝛼) ≈ −∆𝛼 sin 𝛼. Using sin 𝛼 and cos 𝛼 at the Bragg condition for 𝑔 reflection, we only need to include the 𝑔𝑥 and 𝑔𝑧 of the reflection. Considering 𝛼0 , the angle corresponding to Bragg conditions at the maximum and minimum to the intensity profile, the calculation of 𝑠𝑔 is as described in Eq. (6). Because the angles 𝛼 are small, thus for the variation of 𝑠𝑔 : 2𝑘0 𝑠𝑔 ≈ 2𝑘0 𝑔 cos(𝛼 − 𝛼0 ) sin 𝛽
(5)
or 2𝑘0 𝑠𝑔 ≈ 2𝑘0 𝑔∆𝛼 sin 𝛽
(6)
Due to the geometrical factor, some reflections might not reach the maximum intensity. Reflections on the rotation axis must be considered separately and the expression for 𝑠𝑔 is: 2𝑘0 𝑠𝑔 = −𝑔2
(7)
14
These reflections and the ones which do not reach the maximum intensity will not be included in the structure solution and refinement strategy. A detailed mathematical description will be given in a future paper. Calculation of the excitation error of these reflections might be used in a two-beam dynamical case for a few strong reflections to find an approximate thickness of the samples and to calculate the 𝑈𝑔 . DISCUSSION As developed at Stockholm University, the Rotation Electron Diffraction technique (RED) is based on a principle similar to that of single crystal Xray diffractometry. Intensity data are recorded as a set of frames during stepwise rotation about an arbitrary axis. The RED method has been employed successfully for structure analysis using the kinematical interpretation of peak intensities. As a further step to develop the RED method, a precise description of the diffraction geometry throughout the rotation movement has been given. The defined rotation axis was the basic element to extract the intensity data and to calculate the excitation error. 15
Structure refinement should include some kind of dynamical calculation in a few diffraction patterns during the beam tilt. In order to proceed with dynamical calculations, a precise description of the diffraction conditions, refinement of the incident beam 𝑘0 and rotation axis, by a combination of calculations and comparison with experimental data, is required. CONCLUSIONS Defining the excitation error of reflections present in the same diffraction pattern can be used as a key for dynamical interpretation and Bloch wave theory. Because of the high dynamical effect and the same excitation error during all data collection, reflections closer to or in the rotation axis will not be included in the refinement strategy or structure solution (see Eq. (7)). An important conclusion seems to be the use of Kikuchi line patterns during the experimental work. By considering the angle between a Kikuchi line and the beam tilt path, the tilt angular range the reflection is in Bragg conditions can be estimated. The reason behind this is based on the determination of rotation axis and the dynamical effect which has been shown by the rocking ̅ In the development of procedures of this sort, curve of 5̅1̅2 and 512. 16
dynamical refinement remains an important task, together with the improvement in the experimental procedure. We propose, therefore, to develop methods based on a few-beam cases, employing the beam tilt rotation method using either multislice or Bloch wave theory.
ACKNOWLEDGMENTS The authors wish to thank Dr. Peter Oleynikov at the University of Stockholm for help during data collections.
17
REFERENCES ANDERSEN, S., J., ZANDBERG, H. W., JANSEN, J., TRÆHOLT, C. & TUNDAL, U. (1998). The crystal structure of the 𝛽 ′′ phase in Al-Mg-Si alloys. Acta Mater. 49, 3283-3298. BUERGER, M. J. (1964). The Precession Method in X-ray Crystallography. New York: Wiley 18
BUXHUKU, M., HANSEN, V., OLEYNIKOV, P. & GJØNNES, J. (2013). DORSET, D. L. (1995). Comments on the validity of the direct phasing and Fourier methods in electron crystallography. Acta. Cryst. A51,869-879. GJØNNES, J., HANSEN, V., BERG, B.S., RUNDE, P., CHENG, Y. F., GJØNNES, K., DORSET, D. L. & GILMORE, J. (1998). Structure model for the phase AlmFe derived from three-dimensional electron diffraction intensities data collection by a precession technique. Comparison with convergent-beam diffraction. Acta. Cryst. A54, 306-319. KOLB, U., GORRELIK, T., KÜBEL, C., OTTEN, M. & HUBERT, D. (2007). Towards automated diffraction tomography. Ultramicroscopy, 107, 507-513. KOLB, U., GORRELIK, T. & OTTEN, M. (2008). Towards automated diffraction tomography-cell parameter determination. Ultramicroscopy, 108, 763-772. MUGNAIOLI, E., GORELIK, T. & KOLB, U. (2009). Ab initio structure solution from electron diffraction data obtained by a combination of 19
automated
diffraction
tomography
and
precession
technique.
Ultramicroscopy, 109, 758. VINCENT, R. & MIDGLEY, P. A. (1994). Double conical beam-rocking system for measurement of integrated electron-diffraction intensities. Ultramicroscopy 53 271-282. WEIRICH, TH. E., RAMLAU, R., SIMON, A., HOVMÖLLER, S. & ZOU, X. (1996). A crystal structure determinate with 0.02Å accuracy by electron microscopy. Nature (London), 382, 144-146. ZHANG, D., OLEYNIKOV, P., HOVMÖLLER, S. & ZOU, X. D. (2010). Collecting 3D electron diffraction data by rotation method. Z. Kristallogr. 225, 94-102. ZOU, X. D., HOVMÖLLER, S., PARRAS, M., VAKET-REGI, M., GONZALES-CALBET, J. M. & GRENIER, J. C. (1993). The complex perovskite-related superstructure 𝐵𝑎2 𝐹𝑒2 𝑂5 solved by HREM and CIP Acta Cryst., A49, 27-35.
20
Figure captions FIG. 1. Planar section of reciprocal space during beam tilt series taken with RED at: (a) 𝑠𝑔 = 0 for 51̅2̅ and (b) 𝑠𝑔 = 0 for 5̅12 reflection. FIG. 2. Schematic illustration of Laue circle during beam tilts. Laue center circle where 51̅2̅ and 5̅12 reflection are in Bragg conditions are represented respectively by letters 𝑎 and 𝑏.
21
FIG. 3. Kikuchi line patterns for [317] zone axis of a cubic structure. The 𝑎 and 𝑐 show 51̅2̅ and 5̅12 reflection in Bragg conditions due to the intercession of incident beam, 𝑘0 , with the respective Kikuchi lines. FIG. 4. Intensity data available as a set of frames corresponding to the planar section of the reciprocal space. (a) Intensities of some reflections extracted by Multigauge computer program. (b) The arrows mark the reflections which are integrated over several planar section of reciprocal space into integrated peaks. FIG. 5. Relative rocking curves for one Friedel pair of reflections: respectively, blue for 51̅2̅ and red for 5̅12. FIG. 6. Schematic representation of excitation error 𝑠𝑔 and the distance 𝐷 of the reflection from the rotation axis, where XY axis is on the plane and Z axis normal to it.
22
Fig. 1a
Fig. 1b
23
Fig. 2
Fig. 3
Fig. 4a 24
Fig. 4b
25
Fig. 5
26
Fig. 6
27
S0968-4328(17)30033-1 http://dx.doi.org/doi:10.1016/j.micron.2017.06.006 JMIC 2443
To appear in:
Micron
Received date: Revised date: Accepted date:
27-1-2017 9-6-2017 14-6-2017
Please cite this article as: Buxhuku, M., Hansen, V., Gjønnes, J., The measurement of intensities in the Rotation Electron Diffraction Technique.Micron http://dx.doi.org/10.1016/j.micron.2017.06.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
The measurement of intensities in the Rotation Electron Diffraction Technique M. Buxhuku1, V. Hansen1 and J. Gjønnes2 1Department
of Mechanical and Structural Engineering and Material
Science, University of Stavanger, N-4036 Stavanger, Norway 2Department of Physics, University of Oslo, Gaustadalleen 21, N-0371 Oslo,
Norway
Highlits
Intensity data are recorded as a set of frames during stepwise rotation about an arbitrary axis. A precise description of the diffraction geometry throughout the rotation movement has been given. The defined rotation axis was the basic element to extract the intensity data and to calculate the excitation error. To develop the RED method, a precise description of the diffraction geometry throughout the rotation movement has been given. Reflections, which do not reach the maximum intensity, will not be included in the structure solution and refinement strategy
1
ABSTRACT The Rotation Electron Diffraction Technique, recently developed at the University of Stockholm, has been used to acquire three-dimensional electron diffraction data. A mathematical expression to calculate the excitation errors, 𝑠𝑔 , is suggested by considering the diffraction geometry and the rotation axis. In order to plot the rocking curves, diffraction patterns taken from CoP3 cubic structure (a = 7.708Å, space group Im3̅) are used as examples. Intensities of some reflections are derived by Multigauge software during the beam tilt series. Rocking curves for two Friedel pair reflections with different angles from the rotation axis are plotted. Keywords: Rotation axis, Excitation error, Kikuchi lines, Rotation Electron Diffraction Technique INTRODUCTION The successes of X-ray and neutron diffraction methods in structure determination are rooted in the concept of integrated intensities. The intensities, 𝐼ℎ𝑘𝑙 , thus recorded can be taken to represent the square of the 2
structure amplitude, |𝐹ℎ𝑘𝑙 |2 , apart from correction factors. It has taken a long time to establish techniques for electron diffraction that can produce useful and robust experimental intensities (Weirich et al., 1996; Andersen et al., 1998; Zou et al., 1993; Dorset, 1995). This is due to the strong interactions between the electron beam and the atoms in the crystal, which lead to strong dynamical scattering effects. This is especially so in the most common type of electron diffraction patterns, recorded in Selected Area Electron Diffraction (SAED) or micro-diffraction modes, with the incident beam along a major zone axis. Efforts have therefore been made to establish diffraction techniques where multi beams’ dynamical scattering is suppressed to the extent that the recorded intensity data can be used in the structure solution. A basic feature has been to tilt the incident beam off the main zone axis, reducing the number of simultaneously excited beams. The Precession Electron Diffraction technique (PED), invented by Vincent and Midgley (Vincent & Midgley, 1994), is seen as emulating the precession photography in X-ray crystallography (Buerger, 1964). This is achieved by simultaneous precession of the incident beam and the diffracted beams. In
3
practice only the first layer is used; three-dimensional intensity data must be obtained by a combination of several axes (Gjønnes et al., 1998). The recently developed Rotation Electron Diffraction method (RED), employs a different principle (Mugnaioli et al., 2009; Zhang et al., 2010), that might be compared with X-ray single crystal diffractometry. Continuous three-dimensional diffraction intensities are recorded by a combination of a single tilt holder and beam tilt about a common arbitrary axis. Intensity data are available as a set of frames corresponding to the planar section of the reciprocal space. The recorded intensity data are extracted by a computer program, EDT Collect, as maxima or by integrations over several planar sections of reciprocal space. So far, maximum intensities have been interpreted kinematically in structure solution by this technique (Mugnaioli et al., 2009) but with poorer R values. There is thus a need for improvement of the technique and its application in structure solution. The possibility of including dynamical diffraction in the interpretation, and a more precise description of the experiment remains an important task in the development of the method. 4
A key feature will be a precise specification of the path of the incident beam during the rotation experiment, and thus of the diffraction condition of the excited beams. Two illustrations, referred to as two-dimensional representations are shown: a set of Laue circles in the diffraction spot pattern and a line representing the incident beam in a Kikuchi line pattern.
MATERIALS AND METHODS Two-dimensional representation in the RED experiment Two-dimensional patterns, recorded with the incident electron beam along major zone axes, are standard representations in electron diffraction (see Fig. 1). However, recent developments in quantitative electron diffraction crystallography focus on patterns that are recorded with off-axis incident beam direction. In the rotation electron diffraction methods (RED) (Zhang 5
et al., 2010; Kolb et al., 2007; Kolb et al., 2008), three-dimensional electron diffraction data are collected by rotation about an arbitrary axis, by a combination of goniometer tilt motion and beam tilt. For use in structure studies, each detailed analysis of diffraction geometry during the rotation experiment is essential, viz. of the incident beam direction 𝑘0 and the excitation 𝑠𝑔 of diffracted beams. As a basis for the analysis in two steps, we propose a two-dimensional representation within a beam tilt range of a few degrees rotation. We assume in this analysis that a basic structure is known. As the first step for the chosen range, a common normal 𝑍 is selected, i.e. with crystallographic indices 𝑢𝑣𝑤, which will define a zero layer (ZOLZ) of reflections: hkl – and a starting value 𝑘0 (0). A preliminary rotation direction 𝑌 is estimated from recorded patterns, e.g. with methods explained by Buxhuku et al. (Buxhuku et al., 2013). The vector product, 𝑻 = 𝒀𝑥𝒁
6
will define the direction of the change in the incident wave vector as a function of the rotation angle 𝛽: 𝒌0 (𝛽) = 𝒌0 (0) + 𝑛𝛽 𝑻
(1)
where n is a normalization factor. Once 𝒌𝟎 (𝛽) is given, the excitation errors of reflections 𝑔 can be determined from the standard expression 2𝑘𝑠𝑔 = −𝒈𝟐 − 2𝒌𝒈
(2)
For a given rotation axis and incident beam, a calculation of expected beam tilt angles for those reflections in Bragg condition is carried out. The calculated value of 𝛽(𝑔), for 𝒔𝒈 = 0, must be compared with those obtained from the experimental values collected by the RED method, e.g. by a refinement procedure. Diffraction conditions during the RED experiment are visualized in two different ways:
7
In the diffraction pattern by a set of Laue circles in ZOLZ, excitation errors are defined by the distance of the reflections from the circle (see Fig. 2). In the Kikuchi line diagram, which maps the locations of Bragg conditions, 𝑠𝑔 = 0, where 𝒌𝟎 (𝛽) is represented as a line, excitation errors are again represented by a distance from the corresponding Kikuchi line. Beam tilt represented by Kikuchi line patterns We consider the Kikuchi line diagram of [317] zone axis in the crystal (see Fig. 3). The diffraction patterns show that the ℎ − 3ℎ 0 systematic row is close to the rotation axis (Buxhuku et al., 2013), with 1̅30 reflection almost in Bragg condition. The trace of the incident beam, 𝑘0 , during the beam tilt series is represented by a line in the beam diagram: the beam tilt path. When 𝑘0 is at the position of the Kikuchi line’s reflection e.g. 5̅12, the reflection will be in Bragg condition. These diffraction conditions are also shown in Fig. 2 with a Laue circle center at the point a. The Kikuchi diagram representation is well suited to carry out the indexes of the reflections’ excitation in almost the same angular range. 8
Simulated Kikuchi line patterns, of the planar section (see Fig.3), during ±0.75° beam tilt are shown considering the three axes, XYZ, where 𝑌 is consider as rotation axis Y= 13̅0; 𝑍 is the direction of incident beam 𝑘0 , and 𝑋 the tangent projection. The direction of the line representing 𝑘0 (𝛽) is given by vector product as shown in Eq. (1).
The expectation position for a reflection, 𝑔, is obtained for 𝑠𝑔 = 0; the value of 𝛽(𝑔) thus obtained must be compared with experimental values. If the reflection has a 𝑧 − component an additional term will be included. Converting the diffraction patterns into integrated intensities Planar sections of near [317] zone axis have been processed by several stages on Multigauge. Extracted intensities1 of 51̅2̅ and 5̅12 are collected from 35 – 40 diffraction patterns of the same beam tilt series and plotted as
1
The intensities shown on the rocking curve are relative intensities and scaled to maximum 100
9
a function of the beam tilt angle. The strongest intensities of reflections which appeared during data collection are considered as in Bragg conditions. Due to the geometrical or so-called Lorentz factor, some reflections might not reach the maximum intensity. These reflections and the ones on the rotation axis will not be included in the structure solutions and refinements’ strategy. The rotation axis, RED geometry and the beam tilt angles are used to derive the mathematical expression for 𝑠𝑔 (see Eq. (6)). The rocking curves for 51̅2̅ and 5̅12 are plotted as a function of excitation error. The rocking curve for one Friedel pair of reflections is also plotted, with one relative to the other. The curves show interesting features and differences in the maximum intensities of the same excitation error e.g. the maximum intensities for 5̅12 is almost twice higher than 51̅2̅ (see Fig. 5). Maximum intensities have been used and interpreted kinematically, solving a number of structures, but limiting the application for refinement. Reasonable results in terms of structure factors may be associated with integrated intensities collected during beam tilt diffraction pattern. It should be apparent from the present study that the expression for the excitation 10
error, 𝑠𝑔 , derived from the diffraction pattern of a beam tilt series, and the Kikuchi line patterns can be used to determine the refinement strategy. The features on the curves are investigated further by representing the RED experiment by the Kikuchi diagram. The simulated Kikuchi line patterns during beam tilts (see Fig. 6) showed that the rocking curve features of reflections before Bragg conditions are due to the dynamical effects. During the beam tilt series of 5̅12 (see Fig. 7), other reflections, such as 2̅1̅1, 4̅2̅2 , 3̅21, and 1̅30, will appear in Bragg conditions. The 13̅ 0 and 1̅30 reflections are the closest to the rotation axis. Due to the curvature of the Ewald sphere, these two reflections will be in Bragg conditions longer than the other reflections. The simulated Kikuchi line patterns of 51̅2̅ reflection show only 422̅ and 1̅30 reflections occurring in Bragg conditions. It is apparent from the present study that the Kikuchi line patterns can be used to determine the refinement strategy.
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Integrated intensities should be equal to the area under the rocking curve when the extracted intensities of reflections are plotted against the excitation error. CALCULATIONS Plotting the rocking curves and defining 𝐬𝐠 expression The data set in the RED were performed and collected in the parallel electron beam mode. Diffraction patterns were summed up and the intensities of reflections were collected as maximum. Intensities of the reflections from different diffraction patterns were plotted as a function of the beam tilt angle. The Lorentz factor is proportional to the “time” the reflection spends close to Bragg conditions and inversely proportional to the velocity with which the Ewald sphere passes through the diffraction condition. Hence, reflections close to the rotation tilt axis and/or incident beam, 𝑘0 , will be visible on many planar sections of the reciprocal space. In consideration of the RED geometry, simulated Kikuchi line patterns, the
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rotation axis and the beam tilt angles, an expression for the excitation errors, 𝑠𝑔 , is suggested. Referring to Fig. 6, the distance D of reflection, 𝑔, from the rotation axis gives the geometry effect during data collection. The distance D of 𝑔 −reflection is: 𝑔 𝐷 = (𝑘0 cos 𝛼 − ⁄2) sin 𝛽
(3)
Choosing 𝑌 as rotation axis along 𝑍, the optical axis, and 𝑋 perpendicular to the rotation axis and optical axis, the beam in the 𝑋𝑍 plane and 𝑌 component of 𝑔 is irrelevant for the excitation error. In practical terms, the calculation of 𝑠𝑔 for the experimental intensity profiles is: 2𝐾 = −𝟐𝒌𝒈 − 𝑔2 = −2𝑘[𝑔𝑥 + sin 𝛼 𝑔𝑧 cos 𝛼]𝑔2
(4)
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The component of k0 along the Y axis is 𝑘0 sin 𝛼, ∆(sin 𝛼) ≈ ∆𝛼 and ∆(cos 𝛼) ≈ −∆𝛼 sin 𝛼. Using sin 𝛼 and cos 𝛼 at the Bragg condition for 𝑔 reflection, we only need to include the 𝑔𝑥 and 𝑔𝑧 of the reflection. Considering 𝛼0 , the angle corresponding to Bragg conditions at the maximum and minimum to the intensity profile, the calculation of 𝑠𝑔 is as described in Eq. (6). Because the angles 𝛼 are small, thus for the variation of 𝑠𝑔 : 2𝑘0 𝑠𝑔 ≈ 2𝑘0 𝑔 cos(𝛼 − 𝛼0 ) sin 𝛽
(5)
or 2𝑘0 𝑠𝑔 ≈ 2𝑘0 𝑔∆𝛼 sin 𝛽
(6)
Due to the geometrical factor, some reflections might not reach the maximum intensity. Reflections on the rotation axis must be considered separately and the expression for 𝑠𝑔 is: 2𝑘0 𝑠𝑔 = −𝑔2
(7)
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These reflections and the ones which do not reach the maximum intensity will not be included in the structure solution and refinement strategy. A detailed mathematical description will be given in a future paper. Calculation of the excitation error of these reflections might be used in a two-beam dynamical case for a few strong reflections to find an approximate thickness of the samples and to calculate the 𝑈𝑔 . DISCUSSION As developed at Stockholm University, the Rotation Electron Diffraction technique (RED) is based on a principle similar to that of single crystal Xray diffractometry. Intensity data are recorded as a set of frames during stepwise rotation about an arbitrary axis. The RED method has been employed successfully for structure analysis using the kinematical interpretation of peak intensities. As a further step to develop the RED method, a precise description of the diffraction geometry throughout the rotation movement has been given. The defined rotation axis was the basic element to extract the intensity data and to calculate the excitation error. 15
Structure refinement should include some kind of dynamical calculation in a few diffraction patterns during the beam tilt. In order to proceed with dynamical calculations, a precise description of the diffraction conditions, refinement of the incident beam 𝑘0 and rotation axis, by a combination of calculations and comparison with experimental data, is required. CONCLUSIONS Defining the excitation error of reflections present in the same diffraction pattern can be used as a key for dynamical interpretation and Bloch wave theory. Because of the high dynamical effect and the same excitation error during all data collection, reflections closer to or in the rotation axis will not be included in the refinement strategy or structure solution (see Eq. (7)). An important conclusion seems to be the use of Kikuchi line patterns during the experimental work. By considering the angle between a Kikuchi line and the beam tilt path, the tilt angular range the reflection is in Bragg conditions can be estimated. The reason behind this is based on the determination of rotation axis and the dynamical effect which has been shown by the rocking ̅ In the development of procedures of this sort, curve of 5̅1̅2 and 512. 16
dynamical refinement remains an important task, together with the improvement in the experimental procedure. We propose, therefore, to develop methods based on a few-beam cases, employing the beam tilt rotation method using either multislice or Bloch wave theory.
ACKNOWLEDGMENTS The authors wish to thank Dr. Peter Oleynikov at the University of Stockholm for help during data collections.
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REFERENCES ANDERSEN, S., J., ZANDBERG, H. W., JANSEN, J., TRÆHOLT, C. & TUNDAL, U. (1998). The crystal structure of the 𝛽 ′′ phase in Al-Mg-Si alloys. Acta Mater. 49, 3283-3298. BUERGER, M. J. (1964). The Precession Method in X-ray Crystallography. New York: Wiley 18
BUXHUKU, M., HANSEN, V., OLEYNIKOV, P. & GJØNNES, J. (2013). DORSET, D. L. (1995). Comments on the validity of the direct phasing and Fourier methods in electron crystallography. Acta. Cryst. A51,869-879. GJØNNES, J., HANSEN, V., BERG, B.S., RUNDE, P., CHENG, Y. F., GJØNNES, K., DORSET, D. L. & GILMORE, J. (1998). Structure model for the phase AlmFe derived from three-dimensional electron diffraction intensities data collection by a precession technique. Comparison with convergent-beam diffraction. Acta. Cryst. A54, 306-319. KOLB, U., GORRELIK, T., KÜBEL, C., OTTEN, M. & HUBERT, D. (2007). Towards automated diffraction tomography. Ultramicroscopy, 107, 507-513. KOLB, U., GORRELIK, T. & OTTEN, M. (2008). Towards automated diffraction tomography-cell parameter determination. Ultramicroscopy, 108, 763-772. MUGNAIOLI, E., GORELIK, T. & KOLB, U. (2009). Ab initio structure solution from electron diffraction data obtained by a combination of 19
automated
diffraction
tomography
and
precession
technique.
Ultramicroscopy, 109, 758. VINCENT, R. & MIDGLEY, P. A. (1994). Double conical beam-rocking system for measurement of integrated electron-diffraction intensities. Ultramicroscopy 53 271-282. WEIRICH, TH. E., RAMLAU, R., SIMON, A., HOVMÖLLER, S. & ZOU, X. (1996). A crystal structure determinate with 0.02Å accuracy by electron microscopy. Nature (London), 382, 144-146. ZHANG, D., OLEYNIKOV, P., HOVMÖLLER, S. & ZOU, X. D. (2010). Collecting 3D electron diffraction data by rotation method. Z. Kristallogr. 225, 94-102. ZOU, X. D., HOVMÖLLER, S., PARRAS, M., VAKET-REGI, M., GONZALES-CALBET, J. M. & GRENIER, J. C. (1993). The complex perovskite-related superstructure 𝐵𝑎2 𝐹𝑒2 𝑂5 solved by HREM and CIP Acta Cryst., A49, 27-35.
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Figure captions FIG. 1. Planar section of reciprocal space during beam tilt series taken with RED at: (a) 𝑠𝑔 = 0 for 51̅2̅ and (b) 𝑠𝑔 = 0 for 5̅12 reflection. FIG. 2. Schematic illustration of Laue circle during beam tilts. Laue center circle where 51̅2̅ and 5̅12 reflection are in Bragg conditions are represented respectively by letters 𝑎 and 𝑏.
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FIG. 3. Kikuchi line patterns for [317] zone axis of a cubic structure. The 𝑎 and 𝑐 show 51̅2̅ and 5̅12 reflection in Bragg conditions due to the intercession of incident beam, 𝑘0 , with the respective Kikuchi lines. FIG. 4. Intensity data available as a set of frames corresponding to the planar section of the reciprocal space. (a) Intensities of some reflections extracted by Multigauge computer program. (b) The arrows mark the reflections which are integrated over several planar section of reciprocal space into integrated peaks. FIG. 5. Relative rocking curves for one Friedel pair of reflections: respectively, blue for 51̅2̅ and red for 5̅12. FIG. 6. Schematic representation of excitation error 𝑠𝑔 and the distance 𝐷 of the reflection from the rotation axis, where XY axis is on the plane and Z axis normal to it.
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Fig. 1a
Fig. 1b
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Fig. 2
Fig. 3
Fig. 4a 24
Fig. 4b
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Fig. 5
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Fig. 6
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